Z:=SumTools[Hypergeometric][ZeilbergerRecurrence]:


print(`A Quick proof of the Main Result of a 17-page Article that Appeard in the May 2017 issue of the journal Advances in Applied Mathematics`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
print(`In this note we give  one-line, automatic proofs of the main result (Theorem 1.3) of the paper: `):
print(` "Identities involving weighted Catalan, Schroder, and Motkzin paths"`):
print(`by Zhi Chen and Hao Pan, arXiv:1608.02448v2 [math.CO], that appeared recently in Advances in Applied Mathematics`):
print(` v. 86 (May 2017), pp.81-98 `):






print(`The left side of Eq. (1.14), let's call it L1(n), satisfies the recurrence`):
print(``):
print(Z(1/n*binomial(n,k)*binomial(n,k+1)*a^(n-k)*b^k,n,k,L1,0..n)):
print(``):
print(`The right side of Eq. (1.14), let's call it R1(n), satisfies the recurrence`):
print(``):
print(Z(a*binomial(2*k,k)/(k+1)*binomial(n-1,2*k)*(a+b)^(n-1-2*k)*(a*b)^k,n,k,R1,0..n)):
print(``):
print(`Since L1(0)=R1(0), and L1(1)=R1(1) (check!), and the two sequences satisfy the same second-order recurrence this proves (1.14) of the above paper`):



print(``):
print(`------------------------------------------------------------------`):
print(``):


print(`The left side of Eq. (1.15), let's call it L2(n), satisfies the recurrence`):
print(``):
print(Z(b*binomial(n+k,2*k)*binomial(2*k,k)/(k+1)*a^(n-k)*b^k,n,k,L2,0..n)):
print(``):
print(`The right side of Eq. (1.15), let's call it R2(n), satisfies the recurrence`):
print(``):
print(Z((a+b)*1/n*binomial(n,k)*binomial(n,k+1)*b^(n-k)*(a+b)^k,n,k,R2,0..n)):
print(``):
print(`Since L2(0)=R2(0), and L2(1)=R2(1) (check!), and the two sequences satisfy the same second-order recurrence this proves (1.14) of the above paper`):

print(`------------------------------------------------------------------`):
print(``):

print(`The left side of Eq. (1.16), let's call it L3(n), satisfies the recurrence`):
print(``):
print(Z(binomial(n+k,2*k)*binomial(2*k,k)/(k+1)*a^(n-k)*b^k,n,k,L3,0..n)):
print(``):

print(`The right side of Eq. (1.16), let's call it R3(n), satisfies the recurrence`):
print(``):
print(Z((a+b)*binomial(2*k,k)/(k+1)*binomial(n-1,2*k)*(a+2*b)^(n-1-2*k)*(a*b+b^2)^k,n,k,R3,0..n)):
print(``):

print(`Since L3(0)=R3(0), and L3(1)=R3(1) (check!), and the two sequences satisfy the same second-order recurrence this proves (1.16) of the above paper`):

print(``):

print(`------------------------------------------------------------------`):
print(``):

print(`The whole think took`, time(), `seconds. `):


quit: